On 5/21/2012 6:26 PM, Russell Standish wrote:
On Mon, May 21, 2012 at 07:42:01AM -0400, Stephen P. King wrote:
On 5/21/2012 12:33 AM, Russell Standish wrote:
On Sun, May 20, 2012 at 12:06:05PM -0700, meekerdb wrote:
On 5/20/2012 9:27 AM, Stephen P. King wrote:
4) What is the cardinality of "all computations"?
Aleph1.
Actually, it is aleph_0. The set of all computations is
countable. OTOH, the set of all experiences (under COMP) is uncountable
(2^\aleph_0 in fact), which only equals \aleph_1 if the continuity
hypothesis holds.
Hi Russell,
Interesting. Do you have any thoughts on what would follow from
not holding the continuity (Cantor's continuum?) hypothesis?
No - its not my field. My understanding is that the CH has bugger all
impact on quotidian mathematics - the stuff physicists use,
basically. But it has a profound effect on the properties of
transfinite sets. And nobody can decide whether CH should be true or
false (both possibilities produce consistent results).
Hi Russell,
I once thought that consistency, in mathematics, was the indication
of existence but situations like this make that idea a point of
contention... CH and AoC <http://en.wikipedia.org/wiki/Axiom_of_choice>
are two axioms associated with ZF set theory that have lead some people
(including me) to consider a wider interpretation of mathematics. What
if all possible consistent mathematical theories must somehow exist?
Its one reason why Bruno would like to restrict ontology to machines,
or at most integers - echoing Kronecker's quotable "God made the
integers, all else is the work of man".
I understand that, but this choice to restrict makes Bruno's
Idealism even more perplexing to me; how is it that the Integers are
given such special status, especially when we cast aside all possibility
(within our ontology) of the "reality" of the physical world? Without
the physical world to act as a "selection" mechanism for what is "Real",
why the bias for integers? This has been a question that I have tried to
get answered to no avail.
This is the origin of Bruno's claim that COMP entails that physics is
not computable, a corrolory of which is that Digital Physics is
refuted (since DP=>COMP).
Does the symbol "=>" mean "implies"? I get confused ...
Yes, that is the usual meaning. It can also be written (DP or not COMP).
"=>" = "or not"
I am still trying to comprehent that equivalence! BTW, I was
reading a related Wiki article
<http://en.wikipedia.org/wiki/Transposition_%28logic%29> and found the
sentence "the truth of "A implies B" the truth of "Not-B implies
not-A"". That looks familiar... Didn't I write something like that to
Quentin and was rebuffed... I wrote it incorrectly it appears...
Of course in Fortran, it means something entirely different: it
renames a type, much like the typedef statement of C. Sorry, that was
a digression.
That's OK. ;-) I suppose that it is a blessing to be able to "think
in code". ;-)
--
Onward!
Stephen
"Nature, to be commanded, must be obeyed."
~ Francis Bacon
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